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P61 · Extremal configurations · Classic · Applied frontier · Hard

Codebook packing in complex projective space · n = 15

Choose n nonzero vectors in C^d — n points of complex projective space — minimizing the largest normalized Hermitian overlap μ = max |⟨z_i, z_j⟩| / (|z_i||z_j|).

Instanced = 4, n = 15
ObjectiveMinimize the largest coherence μ
Best known, unproven0.447213595the Game of Sloanes leader board (the table of putatively optimal packings maintained by Jasper, King and Mixon)

Formal definition

  • ContainerComplex d-space C^d; an answer is n points of complex projective space CP^{d-1}
  • SubmissionExactly n nonzero complex vectors, each d pairs [re, im]
  • ObjectiveMinimize μ² = max |⟨z_i,z_j⟩|²/(|z_i|²|z_j|²); moduli and norms squared are rational, compared exactly by cross-multiplication
  • ScoringThe record is ceil(μ²·10¹⁸); the page shows μ, rounded up at the ninth decimal
Challenge this record ↗
VERIFIED CONSTRUCTIONpairwise overlap heat of 15 directions; brighter is closer

Getting a feel for it

Where it is used

Complex projective codebooks are quantum measurements (the SIC-POVM family), communication codebooks, and noise-robust data representations. The further apart the codewords, the better they survive noise and erasures.

Where the frontier is

The Game of Sloanes is a public leader board of putatively optimal packings that explicitly invites improvement. Every sub-problem here is chosen from its still-open rows — the ones with a real gap between the best known value and the lower bound.

Source
Current leader

0.997735218

the largest coherence μ

Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
ANSWER FORMAT

How to write your answer

Each complex vector is d pairs [re, im], components written as decimal strings in [-1, 1] with at most nine decimal places. Global phase and nonzero complex scaling represent the same projective point.

Submit vectors: exactly n rows of d pairs [re, im], both decimal strings in [-1, 1]. Each row is a nonzero complex vector; global phase and nonzero complex scaling do not change the answer.

The current leader's answer

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.066666667",
        "0"
      ],
      [
        "0.004444444",
        "0"
      ],
      [
        "0.000296296",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.133333333",
        "0"
      ],
      [
        "0.017777778",
        "0"
      ],
      [
        "0.002370370",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.200000000",
        "0"
      ],
      [
        "0.040000000",
        "0"
      ],
      [
        "0.008000000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.266666667",
        "0"
      ],
      [
        "0.071111111",
        "0"
      ],
      [
        "0.018962963",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.333333333",
        "0"
      ],
      [
        "0.111111111",
        "0"
      ],
      [
        "0.037037037",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.400000000",
        "0"
      ],
      [
        "0.160000000",
        "0"
      ],
      [
        "0.064000000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.466666667",
        "0"
      ],
      [
        "0.217777778",
        "0"
      ],
      [
        "0.101629630",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.533333333",
        "0"
      ],
      [
        "0.284444444",
        "0"
      ],
      [
        "0.151703704",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.600000000",
        "0"
      ],
      [
        "0.360000000",
        "0"
      ],
      [
        "0.216000000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.666666667",
        "0"
      ],
      [
        "0.444444444",
        "0"
      ],
      [
        "0.296296296",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.733333333",
        "0"
      ],
      [
        "0.537777778",
        "0"
      ],
      [
        "0.394370370",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.800000000",
        "0"
      ],
      [
        "0.640000000",
        "0"
      ],
      [
        "0.512000000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.866666667",
        "0"
      ],
      [
        "0.751111111",
        "0"
      ],
      [
        "0.650962963",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.933333333",
        "0"
      ],
      [
        "0.871111111",
        "0"
      ],
      [
        "0.813037037",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 15,
  "d": 4
}

The current leader's answer

{
  "vectors": [
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.066666667",
        "0"
      ],
      [
        "0.004444444",
        "0"
      ],
      [
        "0.000296296",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.133333333",
        "0"
      ],
      [
        "0.017777778",
        "0"
      ],
      [
        "0.002370370",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.200000000",
        "0"
      ],
      [
        "0.040000000",
        "0"
      ],
      [
        "0.008000000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.266666667",
        "0"
      ],
      [
        "0.071111111",
        "0"
      ],
      [
        "0.018962963",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.333333333",
        "0"
      ],
      [
        "0.111111111",
        "0"
      ],
      [
        "0.037037037",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.400000000",
        "0"
      ],
      [
        "0.160000000",
        "0"
      ],
      [
        "0.064000000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.466666667",
        "0"
      ],
      [
        "0.217777778",
        "0"
      ],
      [
        "0.101629630",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.533333333",
        "0"
      ],
      [
        "0.284444444",
        "0"
      ],
      [
        "0.151703704",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.600000000",
        "0"
      ],
      [
        "0.360000000",
        "0"
      ],
      [
        "0.216000000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.666666667",
        "0"
      ],
      [
        "0.444444444",
        "0"
      ],
      [
        "0.296296296",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.733333333",
        "0"
      ],
      [
        "0.537777778",
        "0"
      ],
      [
        "0.394370370",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.800000000",
        "0"
      ],
      [
        "0.640000000",
        "0"
      ],
      [
        "0.512000000",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.866666667",
        "0"
      ],
      [
        "0.751111111",
        "0"
      ],
      [
        "0.650962963",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "0.933333333",
        "0"
      ],
      [
        "0.871111111",
        "0"
      ],
      [
        "0.813037037",
        "0"
      ]
    ],
    [
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ],
      [
        "1.000000000",
        "0"
      ]
    ]
  ]
}

Submit vectors: exactly n rows of d pairs [re, im], both decimal strings in [-1, 1]. Each row is a nonzero complex vector; global phase and nonzero complex scaling do not change the answer. · Verifier v1.0.0

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